Properties

 Label 1260.2.a.j Level $1260$ Weight $2$ Character orbit 1260.a Self dual yes Analytic conductor $10.061$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$1260 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1260.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$10.0611506547$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{5} + q^{7} + O(q^{10})$$ $$q + q^{5} + q^{7} - 4 q^{13} + 6 q^{17} + 2 q^{19} + 6 q^{23} + q^{25} + 2 q^{31} + q^{35} + 2 q^{37} + 6 q^{41} - 4 q^{43} + q^{49} - 6 q^{53} + 12 q^{59} - 10 q^{61} - 4 q^{65} - 4 q^{67} + 12 q^{71} - 4 q^{73} + 8 q^{79} + 12 q^{83} + 6 q^{85} + 6 q^{89} - 4 q^{91} + 2 q^{95} + 8 q^{97} + O(q^{100})$$

Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 0 0 1.00000 0 1.00000 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$1$$
$$5$$ $$-1$$
$$7$$ $$-1$$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1260.2.a.j yes 1
3.b odd 2 1 1260.2.a.d 1
4.b odd 2 1 5040.2.a.w 1
5.b even 2 1 6300.2.a.h 1
5.c odd 4 2 6300.2.k.j 2
7.b odd 2 1 8820.2.a.h 1
12.b even 2 1 5040.2.a.e 1
15.d odd 2 1 6300.2.a.i 1
15.e even 4 2 6300.2.k.i 2
21.c even 2 1 8820.2.a.u 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1260.2.a.d 1 3.b odd 2 1
1260.2.a.j yes 1 1.a even 1 1 trivial
5040.2.a.e 1 12.b even 2 1
5040.2.a.w 1 4.b odd 2 1
6300.2.a.h 1 5.b even 2 1
6300.2.a.i 1 15.d odd 2 1
6300.2.k.i 2 15.e even 4 2
6300.2.k.j 2 5.c odd 4 2
8820.2.a.h 1 7.b odd 2 1
8820.2.a.u 1 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1260))$$:

 $$T_{11}$$ $$T_{17} - 6$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$-1 + T$$
$7$ $$-1 + T$$
$11$ $$T$$
$13$ $$4 + T$$
$17$ $$-6 + T$$
$19$ $$-2 + T$$
$23$ $$-6 + T$$
$29$ $$T$$
$31$ $$-2 + T$$
$37$ $$-2 + T$$
$41$ $$-6 + T$$
$43$ $$4 + T$$
$47$ $$T$$
$53$ $$6 + T$$
$59$ $$-12 + T$$
$61$ $$10 + T$$
$67$ $$4 + T$$
$71$ $$-12 + T$$
$73$ $$4 + T$$
$79$ $$-8 + T$$
$83$ $$-12 + T$$
$89$ $$-6 + T$$
$97$ $$-8 + T$$